Scaling Equation Between Response Spectrum and Energy Spectrum

Abstract

In this paper, the scaling equation between the pseudo velocity response spectrum and the energy spectrum is proposed. Utilizing random vibration theory, the maximum response of SDOF is given as the function of power spectrum density of main part of earthquake motion. Whereas, energy spectrum is equal to Fourier amplitude spectrum smoothed by spectral window, which is clearly pointed out by Kuwamura, et al. Finally, the equation between response spectrum and energy spectrum is derived, by making connection between Fourier amplitude spectrum and power spectrum density considering the nonstationarity of earthquake motion. The proposed equation is verified not only using the artificial earthquake motions but also using the observed strong ground motions of both interplate earthquakes and inland crustal earthquakes. The equation between the peak ground acceleration/velocity and response spectrum is also proposed, which is the application of the method shown in this paper.

Spectrum Conversion Factor and its Application for Ground Motion Simulation

“…(6) underestimate the SCF in 0.1 -5.0 Hz. The three previous studies (Akiyama and Kitamura 2006, Miyake 2006, and Okano et al 2009) assumed that the duration of strong motion is proportional to the total energy of the ground motion. Therefore, those studies modeled the V E (square root of the input energy) by the square root of the duration of strong motion.…”

“…All of the standard errors s were assumed to be constant along σ. In Fig.6, regression results when the SCF are expressed by the following form (as is done in Akiyama and Kitamura 2006, Miyake 2006, and Okano et al 2009 …”

“…The most recent research for S v -V E ratio would be Okano et al (2009). They derived a theoretical S v -V E ratio using a relationship between the power spectrum and response spectrum for white noise.…”

Earthquake response prediction of seismically isolated buildings based on the response spectrum method considering changes in characteristics of lead rubber bearings due to repeated deformations

“…Table 3 shows the start and end times of the principal motion of the nine LPGMDs and the duration of the principal motion interval. The method to determine the principal motion duration was based on the method of Okano et al., 10 as shown below. The cumulative squared value of the acceleration waveform was considered to represent the supplied energy, and the tangential slope of the energy supply rate at the time when the energy supply rate was 50%, was used as the reference.…”